**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

p. 279 -286

M. A. Kamal and O. A. Elmnophy

**Abstract.** Let R be a ring. A right R-module M is called quasi-principally injective if it is
M-principally injective.In this paper, we give some characterizations and properties of
principally injective modules, which generalize results of Nicholson and Yousif. For a
quasi-principally injective module M, we show: 1. For isomorphic submodules H, K of
M, we have SH = SK, where S is the endomorphism ring of M. 2. M has
(PC_{2}), and consequently has (PC_{3}). We characterize when a direct sum of
P-extending modules is P-extending, and when a direct sum of a P-extending module
and a semi-simple module is P-extending. We also characterize when a direct
sum of FP-extending modules is FP-extending. Finally, we discuss when a
direct sum of P-extending modules with relatively EC-injective is P-extending.

**Keywords**:
Principally injective modules, extending modules.
**AMS Subject classification:** 16D50, 16D70, 16D80.

Institute of Applied Mathematics

Faculty of Mathematics, Physics and Informatics

Comenius University

842 48 Bratislava, Slovak Republic

Telephone: + 421-2-60295755 Fax: + 421-2-65425882

e-Mail: amuc@fmph.uniba.sk Internet: www.iam.fmph.uniba.sk/amuc

© Copyright 2005, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE